Математика и авиация

Mathematics in Aviation
Maintenance
Many examples of using mathematical principles
by the aviation mechanic are available. Tolerances
in turbine engine components are critical, making it
necessary to measure within a ten-thousandth of an
inch. Because of these close tolerances, it is important
that the aviation mechanic be able to make accurate
measurements and mathematical calculations. An aviation mechanic working on aircraft fuel systems will
also use mathematical principles to calculate volumes
and capacities of fuel tanks. The use of fractions and
surface area calculations are required to perform sheet
metal repair on aircraft structures.
Whole Numbers
Whole numbers are the numbers 0, 1, 2, 3, 4, 5, and
so on.
Addition of Whole Numbers
Addition is the process in which the value of one
number is added to the value of another. The result is
called the sum. When working with whole numbers,
it is important to understand the principle of the place
value. The place value in a whole number is the value
of the position of the digit within the number. For
example, in the number 512, the 5 is in the hundreds
column, the 1 is in the tens column, and the 2 is in the
ones column. The place values of three whole numbers
are shown in Figure 1-1.
Tens
Ones
Hundreds
Thousands
Ten
Thousands
Mathematics is woven into many areas of everyday life.
Performing mathematical calculations with success
requires an understanding of the correct methods and
procedures, and practice and review of these principles.
Mathematics may be thought of as a set of tools. The
aviation mechanic will need these tools to successfully complete the maintenance, repair, installation,
or certification of aircraft equipment.
Place Value
3
5
2
6
9
7
4
9
35 shown as
269 shown as
12,749 shown as
1
2
Figure 1-1. Example of place values of whole numbers.
When adding several whole numbers, such as 4,314,
122, 93,132, and 10, align them into columns according
to place value and then add.
4,314
122
93,132
+ 10
97,578 This is the sum of the four
whole numbers.
Subtraction of Whole Numbers
Subtraction is the process in which the value of one
number is taken from the value of another. The answer
is called the difference. When subtracting two whole
numbers, such as 3,461 from 97,564, align them into
columns according to place value and then subtract.
97,564
–3,461
94,103 This is the difference of the two
whole numbers.
Multiplication of Whole Numbers
Multiplication is the process of repeated addition. For
example, 4 × 3 is the same as 4 + 4 + 4. The result is
called the product.
Example: How many hydraulic system filters are in
the supply room if there are 35 cartons and each carton
contains 18 filters?
1-1
18
× 35
90
54
630
Therefore, there are 630 filters in the supply room.
Division of Whole Numbers
Division is the process of finding how many times one
number (called the divisor) is contained in another
number (called the dividend). The result is the quotient,
and any amount left over is called the remainder.
quotient
divisor dividend
Example: 218 landing gear bolts need to be divided
between 7 aircraft. How many bolts will each aircraft
receive?
31
7 218
− 21
8
−7
1
The solution is 31 bolts per aircraft with a remainder
of 1 bolt left over.
Fractions
A fraction is a number written in the form N⁄D where N
is called the numerator and D is called the denominator.
The fraction bar between the numerator and denominator shows that division is taking place.
17 2 5
Some examples of fractions are:
18 3 8
The denominator of a fraction cannot be a zero. For
example, the fraction 2⁄ 0 is not allowed. An improper
fraction is a fraction in which the numerator is equal
to or larger than the denominator. For example, 4⁄4 or
15⁄ are examples of improper fractions.
8
Finding the Least Common Denominator
To add or subtract fractions, they must have a common
denominator. In math, the least common denominator
(LCD) is commonly used. One way to find the LCD
is to list the multiples of each denominator and then
choose the smallest one that they have in common.
Example: Add 1⁄5 + 1⁄10 by finding the least common
denominator.
1-2
Multiples of 5 are: 5, 10, 15, 20, 25, and on. Multiples
of 10 are: 10, 20, 30, 40, and on. Notice that 10, 20,
and 30 are in both lists, but 10 is the smallest or least
common denominator (LCD). The advantage of finding the LCD is that the final answer is more likely to
be in lowest terms.
A common denominator can also be found for any
group of fractions by multiplying all of the denominators together. This number will not always be the LCD,
but it can still be used to add or subtract fractions.
Example: Add 2⁄3 + 3⁄5 + 4⁄7 by finding a common
denominator.
A common denominator can be found by multiplying
the denominators 3 × 5 × 7 to get 105.
2
3
3
5
4
7
70
105
63
105
60
105
193
105
1
88
105
Addition of Fractions
In order to add fractions, the denominators must be the
same number. This is referred to as having “common
denominators.”
Example: Add 1⁄7 to 3⁄7
1
7
1+3
7
3
7
4
7
If the fractions do not have the same denominator, then
one or all of the denominators must be changed so that
every fraction has a common denominator.
Example: Find the total thickness of a panel made from
3⁄ -inch thick aluminum, which has a paint coating that
32
is 1⁄64-inch thick. To add these fractions, determine a
common denominator. The least common denominator
for this example is 1, so only the first fraction must be
changed since the denominator of the second fraction
is already 64.
3
32
1
64
3×2
32 × 2
1
64
6
64
1
64
6+1
64
7
64
Therefore, 7⁄64 is the total thickness.
Subtraction of Fractions
In order to subtract fractions, they must have a common denominator.
Example: Subtract 2⁄17 from 10⁄17
10 − 2
17 17
10 − 2
17
8
17
If the fractions do not have the same denominator, then
one or all of the denominators must be changed so that
every fraction has a common denominator.
Example: The tolerance for rigging the aileron droop of
an airplane is 7⁄8 inch ± 1⁄5 inch. What is the minimum
droop to which the aileron can be rigged? To subtract
these fractions, first change both to common denominators. The common denominator in this example
is 40. Change both fractions to 1⁄40, as shown, then
subtract.
7
1
−
8
5
7×5
1×8
8×5 − 5×8
35
8
40 − 40
35 − 8
40
27
40
3 –5
8
7
5 ––
16
Figure 1-2. Center hole of the plate.
Therefore, 27⁄40 is the minimum droop.
Multiplication of Fractions
Multiplication of fractions does not require a common
denominator. To multiply fractions, first multiply the
numerators. Then, multiply the denominators.
Example:
3 7
1
×
×
5 8
2
3×7×1
5×8×2
21
80
The use of cancellation when multiplying fractions is
a helpful technique which divides out or cancels all
common factors that exist between the numerators and
denominators. When all common factors are cancelled
before the multiplication, the final product will be in
lowest terms.
Example:
14 × 3
15
7
2
1
14 × 3
15
7
5
1
2×1
5×1
2
5
Division of Fractions
Division of fractions does not require a common
denominator. To divide fractions, first change the division symbol to multiplication. Next, invert the second
fraction. Then, multiply the fractions.
Example: Divide 7⁄8 by 4⁄3
7
8
4
3
7 3
×
8 4
7×3
8×4
21
32
Example: In Figure 1-2, the center of the hole is in the
center of the plate. Find the distance that the center
of the hole is from the edges of the plate. To find the
answer, the length and width of the plate should each
be divided in half. First, change the mixed numbers to
improper fractions:
5 7⁄16 inches = 87⁄16 inches
3 5⁄8 inches = 29⁄8 inches
Then, divide each improper fraction by 2 to find the
center of the plate.
87
16
2
1
87 1
×
16 2
87
inches
32
29
8
2
1
29 1
×
8
2
29 inches
16
Finally, convert each improper fraction to a mixed
number:
87
32
87 32
2
23 inches
32
29
16
29 16
1
13 inches
16
Therefore, the distance to the center of the hole from
each of the plate edges is 2 23⁄32 inches and 1 13⁄16
inches.
Reducing Fractions
A fraction needs to be reduced when it is not in “lowest
terms.” Lowest terms means that the numerator and
denominator do not have any factors in common. That
is, they cannot be divided by the same number (or factor). To reduce a fraction, determine what the common
factor(s) are and divide these out of the numerator and
denominator. For example when both the numerator
and denominator are even numbers, they can both be
divided by 2.
Example: The total travel of a jackscrew is 13⁄16 inch. If
the travel in one direction from the neutral position is
7⁄ inch, what is the travel in the opposite direction?
16
13
7
−
16 16
13 − 7
16
6
16
1-3
The fraction 6⁄16 is not in lowest terms because the
numerator (6) and the denominator (16) have a common factor of 2. To reduce 6⁄16, divide the numerator
and the denominator by 2. The final reduced fraction
is 3⁄8 as shown below.
6
6÷2
3
16 ÷ 2
16
8
Therefore, the travel in the opposite direction is 3⁄8
inch.
Subtraction of Mixed Numbers
To subtract mixed numbers, find a common denominator for the fractions. Subtract the fractions from
each other (it may be necessary to borrow from the
larger whole number when subtracting the fractions).
Subtract the whole numbers from each other. The final
step is to combine the final whole number with the
final fraction.
Therefore, the grip length of the bolt is 113⁄16 inches.
(Note: The value for the overall length of the bolt was
given in the example, but it was not needed to solve
the problem. This type of information is sometimes
referred to as a “distracter” because it distracts from
the information needed to solve the problem.)
The Decimal Number System
The Origin and Definition
The number system that we use every day is called
the decimal system. The prefix in the word decimal is
a Latin root for the word “ten.” The decimal system
probably had its origin in the fact that we have ten
fingers (or digits). The decimal system has ten digits:
0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. It is a base 10 system
and has been in use for over 5,000 years. A decimal is
a number with a decimal point. For example, 0.515,
.10, and 462.625 are all decimal numbers. Like whole
numbers, decimal numbers also have place value. The
place values are based on powers of 10, as shown in
Figure 1-4.
Example: What is the length of the grip of the bolt
shown in Figure 1-3? The overall length of the bolt
is 3 1⁄2 inches, the shank length is 3 1⁄8 inches, and the
threaded portion is 1 5⁄16 inches long. To find the grip,
subtract the length of the threaded portion from the
length of the shank.
Place Value
3 1⁄8 inches – 1 5⁄16 inches = grip length
To subtract, start with the fractions. Borrowing will be
necessary because 5⁄16 is larger than 1⁄8 (or 2⁄16). From
the whole number 3, borrow 1, which is actually 16⁄16.
After borrowing, the first mixed number will now be
1-4
1,623,051
1
6
2
3
0
5
1
0.0531
32.4
3
Ten Thousandths
13
12
13
16
Thousandths
6
1
Hundredths
4
12
18
5
−1
16
16
Tenths
9
12
2
Ones
6
2 − 5
1
16
16
Tens
1
3
3
Hundreds
3
1
3
2
(4 + 2) +
4
3
4
1
feet of cargo room.
7
12
4
1
5
−1
8
16
Thousands
Example: The cargo area behind the rear seat of a small
airplane can handle solids that are 4 3⁄4 feet long. If the
rear seats are removed, then 2 1⁄3 feet is added to the
cargo area. What is the total length of the cargo area
when the rear seats are removed?
3
Ten Thousands
Addition of Mixed Numbers
To add mixed numbers, add the whole numbers
together. Then add the fractions together by finding a
common denominator. The final step is to add the sum
of the whole numbers to the sum of the fractions for
the final result.
2 18⁄16 . (Because, 31⁄8 = 3 2⁄16 = 2 + 1 + 2⁄16 = 2 + 16⁄16
+ 2⁄16 = 218⁄16.)
Hundred Thousands
A mixed number is a combination of a whole number
and a fraction.
Millions
Mixed Numbers
Figure 1-3. Bolt dimensions.
0
0
5
3
1
2
4
Figure 1-4. Place values.
Addition of Decimal Numbers
To add decimal numbers, they must first be arranged
so that the decimal points are aligned vertically and
according to place value. That is, adding tenths with
tenths, ones with ones, hundreds with hundreds, and
so forth.
2.34 Ohms
37.5 Ohms
Example: Find the total resistance for the circuit diagram shown in Figure 1-5. The total resistance of a
series circuit is equal to the sum of the individual resistances. To find the total resistance, RT, the individual
resistances are added together.
.09 Ohms
RT = 2.34 + 37.5 + .09
Figure 1-5. Circuit diagram.
Arrange the resistance values in a vertical column so
that the decimal points are aligned and then add.
2.34
37.5
+ .09
39.93
Therefore, the total resistance, RT = 39.93 ohms.
Subtraction of Decimal Numbers
To subtract decimal numbers, they must first be
arranged so that the decimal points are aligned vertically and according to place value. That is, subtracting
tenths from tenths, ones from ones, hundreds from
hundreds, and so forth.
Example: A series circuit containing two resistors has
a total resistance (RT) of 37.272 ohms. One of the
resistors (R1) has a value of 14.88 ohms. What is the
value of the other resistor (R2)?
R2 = RT – R1 = 37.272 – 14.88
Arrange the decimal numbers in a vertical column so
that the decimal points are aligned and then subtract.
37.272
− 14.88
22.392
Therefore, the second resistor, R2 = 22.392 ohms.
Multiplication of Decimal Numbers
To multiply decimal numbers, vertical alignment of
the decimal point is not required. Instead, align the
numbers to the right in the same way as whole numbers
are multiplied (with no regard to the decimal points or
M
place values) and then multiply. The last step is to place
the decimal point in the correct place in the answer.
To do this, “count” the number of decimal places in
each of the numbers, add the total, and then “give” that
number of decimal places to the result.
Example: To multiply 0.2 × 6.03, arrange the numbers
vertically and align them to the right. Multiply the
numbers, ignoring the decimal points for now.
6.03
× 0.2
1206 (ignore the decimal points, for now)
After multiplying the numbers, count the total number
of decimal places in both numbers. For this example,
6.03 has 2 decimal places and 0.2 has 1 decimal place.
Together there are a total of 3 decimal places. The
decimal point for the answer will be placed 3 decimal
places from the right. Therefore, the answer is 1.206.
6.03
× 0.2
1.206
2 decimal places
1 decimal place
3 decimal places
Example: Using the formula Watts = Amperes × Voltage, what is the wattage of an electric drill that uses
9.45 amperes from a 120 volt source? Align the numbers to the right and multiply.
After multiplying the numbers, count the total number
of decimal places in both numbers. For this example,
9.45 has 2 decimal places and 120 has no decimal place.
Together there are 2 decimal places. The decimal point
for the answer will be placed 2 decimal places from
1-5
the right. Therefore, the answer is 1,134.00 watts, or
1,134 watts.
9.45
× 120
000
1890
945
1,134.00
2 decimal places
no decimal places
2 decimal places
Division of Decimal Numbers
Division of decimal numbers is performed the
same way as whole numbers, unless the divisor is a
decimal.
quotient
divisor dividend
When the divisor is a decimal, it must be changed to
a whole number before dividing. To do this, move the
decimal in the divisor to the right until there are no
decimal places. At the same time, move the decimal
point in the dividend to the right the same number of
places. Then divide. The decimal in the quotient will
be placed directly above the decimal in the dividend.
Example: Divide 0.144 by 0.12
1.2
.12 0.144 = 12. 14.4
12
24
24
0
Move the decimal in the divisor (0.12) two places
to the right. Next move the decimal in the dividend
(0.144) two places to the right. Then divide. The result
is 1.2.
Example: The wing area of an airplane is 262.6 square
feet and its span is 40.4 feet. Find the mean chord of its
wing using the formula: Area ÷ span = mean chord.
6.5
40.4 262.6 = 404. 2626.0
2424
2020
2020
0
Move the decimal in the divisor (40.4) one place to the
right. Next move the decimal in the dividend (262.6)
one place to the right. Then divide. The mean chord
length is 6.5 feet.
1-6
Rounding Off Decimal Numbers
Occasionally, it is necessary to round off a decimal
number to some value that is practical to use.
For example, a measurement is calculated to be
29.4948 inches. To use this measurement, we can use
the process of “rounding off.” A decimal is “rounded
off” by keeping the digits for a certain number of places
and discarding the rest. The degree of accuracy desired
determines the number of digits to be retained. When
the digit immediately to the right of the last retained
digit is 5 or greater, round up by 1. When the digit
immediately to the right of the last retained digit is less
than 5, leave the last retained digit unchanged.
Example: An actuator shaft is 2.1938 inches in diameter. Round to the nearest tenth.
The digit in the tenths column is a 1. The digit to the
right of the 1 is a 9. Since 9 is greater than or equal to
5, “round up” the 1 to a 2. Therefore, 2.1938 rounded
to the nearest tenth is 2.2.
Example: The outside diameter of a bearing is 2.1938
inches. Round to the nearest hundredth.
The digit in the hundredths column is a 9. The digit
to the right of the 9 is a 3. Since 3 is less than 5, do
not round up the 9. Therefore, 2.1938 to the nearest
hundredth is 2.19.
Example: The length of a bushing is 2.1938 inches.
Round to the nearest thousandth.
The digit in the thousandths column is a 3. The digit
to the right of the 3 is an 8. Since 8 is greater than or
equal to 5, “round up” the 3 to a 4. Therefore, 2.1938
to the nearest thousandth is 2.194.
Converting Decimal Numbers to Fractions
To change a decimal number to a fraction, “read” the
decimal, and then write it into a fraction just as it is
read as shown below.
Example: One oversized rivet has a diameter of 0.52
inches. Convert 0.52 to a fraction. The decimal 0.52
is read as “fifty-two hundredths.”
Therefore,
0.52 = 52
100
“fifty-two”
which can be reduced to 13
25
“hundredths”
A dimension often appears in a maintenance manual
or on a blueprint as a decimal instead of a fraction. In
order to use the dimension, it may need to be converted
to a fraction. An aviation mechanic frequently uses a
steel rule that is calibrated in units of 1⁄64 of an inch.
To change a decimal to the nearest equivalent common
fraction, multiply the decimal by 64. The product of the
decimal and 64 will be the numerator of the fraction
and 64 will be the denominator. Reduce the fraction,
if needed.
Example: The width of a hex head bolt is 0.3123
inches. Convert the decimal 0.3123 to a common
fraction to decide which socket would be the best fit
for the bolt head. First, multiply the 0.3123 decimal
by 64:
0.3123 × 64 = 19.9872
Next, round the product to the nearest whole number:
19.98722 ≈ 20.
Use this whole number (20) as the numerator and 64
as the denominator: 20⁄64 .
Now, reduce
20⁄
64
to
5⁄ .
16
Therefore, the correct socket would be the 5⁄16 inch
socket ( 20⁄64 reduced).
Example: When accurate holes of uniform diameter
are required for aircraft structures, they are first drilled
approximately 1⁄64 inch undersized and then reamed to
the final desired diameter. What size drill bit should
be selected for the undersized hole if the final hole is
reamed to a diameter of 0.763 inches? First, multiply
the 0.763 decimal by 64.
0.763 × 64 = 48.832
Example:
.5
1
2 = 1 ÷ 2 = 2 1.0
1.0
0
.375
3
8
3.000
=
=
3
÷
8
8
– 24
60
– 56
40
– 40
0
49
1
48
3
64 − 64 = 64 = 4
Therefore, a 3⁄4-inch drill bit should be used for the
initial undersized holes.
Converting Fractions to Decimals
To convert any fraction to a decimal, simply divide
the top number (numerator) by the bottom number
(denominator). Every fraction will have an approximate decimal equivalent.
3
Therefore, 8 = .375
Calculator tip: Numerator (top number) ÷ Denominator (bottom number) = the decimal equivalent of the
fraction.
Some fractions when converted to decimals produce
a repeating decimal.
Example:
.33
1
3 = 1 ÷ 3 = 3 1.00 = .3 or .33
This decimal can be
–9
represented with bar, or can
10
be rounded. (A bar indicates
–9
that the number(s) beneath it
1
are repeated to infinity.)
Other examples of repeating decimals:
.212121… = .21
Next, round the product to the nearest whole number:
48.832 ≈ 49.
Use this number (49) as the numerator and 64 as the
denominator: 49⁄64 is the closest fraction to the final
reaming diameter of 0.763 inches. To determine the
drill size for the initial undersized hole, subtract 1⁄64
inch from the finished hole size.
1
Therefore, 2 = .5
.6666… = .7 or .67
.254254… = .254
Decimal Equivalent Chart
Figure 1-6 (on the next page) is a fraction to decimal
to millimeter equivalency chart. Measurements starting
at 1⁄64 inch and up to 23 inches have been converted to
decimal numbers and to millimeters.
Ratio
A ratio is the comparison of two numbers or quantities.
A ratio may be expressed in three ways: as a fraction,
with a colon, or with the word “to.” For example, a gear
ratio of 5:7 can be expressed as any of the following:
5⁄
7
or 5:7 or 5 to 7
1-7
fraction decimal
1/64
1/32
3/64
1/16
5/64
3/32
7/64
1/8
9/64
5/32
11/64
5/16
13/64
7/32
15/64
1/4
17/64
9/32
19/64
5/16
21/64
11/32
23/64
3/8
25/64
13/32
27/64
7/16
29/64
15/32
31/64
1/2
33/64
17/32
35/64
39341
37/64
19/32
39/64
5/8
41/64
21/32
43/64
11/16
45/64
23/32
47/64
3/4
49/64
25/32
51/64
13/16
53/64
27/32
55/64
7/8
57/64
29/32
59/64
15/16
61/64
31/32
63/64
1
0.015
0.031
0.046
0.062
0.078
0.093
0.109
0.125
0.140
0.156
0.171
0.187
0.203
0.218
0.234
0.25
0.265
0.281
0.296
0.312
0.328
0.343
0.359
0.375
0.390
0.406
0.421
0.437
0.453
0.468
0.484
0.5
0.515
0.531
0.546
0.562
0.578
0.593
0.609
0.625
0.640
0.656
0.671
0.687
0.703
0.718
0.734
0.75
0.765
0.781
0.796
0.812
0.828
0.843
0.859
0.875
0.890
0.906
0.921
0.937
0.953
0.968
0.984
1
mm
0.396
0.793
1.190
1.587
1.984
2.381
2.778
3.175
3.571
3.968
4.365
4.762
5.159
5.556
5.953
6.35
6.746
7.143
7.540
7.937
8.334
8.731
9.128
9.525
9.921
10.318
10.715
11.112
11.509
11.906
12.303
12.7
13.096
13.493
13.890
14.287
14.684
15.081
15.478
15.875
16.271
16.668
17.065
17.462
17.859
18.256
18.653
19.05
19.446
19.843
20.240
20.637
21.034
21.431
21.828
22.225
22.621
23.018
23.415
23.812
24.209
24.606
25.003
25.4
fraction decimal
1 1/64
1 1/32
1 3/64
1 1/16
1 5/64
1 3/32
1 7/64
1 1/8
1 9/64
1 5/32
1 11/64
1 3/16
1 13/64
1 7/32
1 15/64
1 1/4
1 17/64
1 9/32
1 19/64
1 5/16
1 21/64
1 11/32
1 23/64
1 3/8
1 25/64
1 13/32
1 27/64
1 7/16
1 29/64
1 15/32
1 31/64
1 1/2
1 33/64
1 17/32
1 35/64
1 9/16
1 37/64
1 19/32
1 39/64
1 5/8
1 41/64
1 21/32
1 43/64
1 11/16
1 45/64
1 23/32
1 47/64
1 3/4
1 49/64
1 25/32
1 51/64
1 13/16
1 53/64
1 27/32
1 55/64
1 7/8
1 57/64
1 29/32
1 59/64
1 15/16
1 61/64
1 31/32
1 63/64
2
1.015
1.031
1.046
1.062
1.078
1.093
1.109
1.125
1.140
1.156
1.171
1.187
1.203
1.218
1.234
1.25
1.265
1.281
1.296
1.312
1.328
1.343
1.359
1.375
1.390
1.406
1.421
1.437
1.453
1.468
1.484
1.5
1.515
1.531
1.546
1.562
1.578
1.593
1.609
1.625
1.640
1.656
1.671
1.687
1.703
1.718
1.734
1.75
1.765
1.781
1.796
1.812
1.828
1.843
1.859
1.875
1.890
1.906
1.921
1.937
1.953
1.968
1.984
2
mm
25.796
26.193
26.590
26.987
27.384
27.781
28.178
28.575
28.971
29.368
29.765
30.162
30.559
30.956
31.353
31.75
32.146
32.543
32.940
33.337
33.734
34.131
34.528
34.925
35.321
35.718
36.115
36.512
36.909
37.306
37.703
38.1
38.496
38.893
39.290
39.687
40.084
40.481
40.878
41.275
41.671
42.068
42.465
42.862
43.259
43.656
44.053
44.45
44.846
45.243
45.640
46.037
46.434
46.831
47.228
47.625
48.021
48.418
48.815
49.212
49.609
50.006
50.403
50.8
Figure 1-6. Fractions, decimals, and millimeters.
1-8
fraction decimal
2 1/64
2 1/32
2 3/64
2 1/16
2 5/64
2 3/32
2 7/64
2 1/8
2 9/64
2 5/32
2 11/64
2 3/16
2 13/64
2 7/32
2 15/64
2 1/4
2 17/64
2 9/32
2 19/64
2 5/16
2 21/64
2 11/32
2 23/64
2 3/8
2 25/64
2 13/32
2 27/64
2 7/16
2 29/64
2 15/32
2 31/64
2 1/2
2 33/64
2 17/32
2 35/64
2 9/16
2 37/64
2 19/32
2 39/64
2 5/8
2 41/64
2 21/32
2 43/64
2 11/16
2 45/64
2 23/32
2 47/64
2 3/4
2 49/64
2 25/32
2 51/64
2 13/16
2 53/64
2 27/32
2 55/64
2 7/8
2 57/64
2 29/32
2 59/64
2 15/16
2 61/64
2 31/32
2 63/64
3
2.015
2.031
2.046
2.062
2.078
2.093
2.109
2.125
2.140
2.156
2.171
2.187
2.203
2.218
2.234
2.25
2.265
2.281
2.296
2.312
2.328
2.343
2.359
2.375
2.390
2.406
2.421
2.437
2.453
2.468
2.484
2.5
2.515
2.531
2.546
2.562
2.578
2.593
2.609
2.625
2.640
2.656
2.671
2.687
2.703
2.718
2.734
2.75
2.765
2.781
2.796
2.812
2.828
2.843
2.859
2.875
2.890
2.906
2.921
2.937
2.953
2.968
2.984
3
mm
51.196
51.593
51.990
52.387
52.784
53.181
53.578
53.975
54.371
54.768
55.165
55.562
55.959
56.356
56.753
57.15
57.546
57.943
58.340
58.737
59.134
59.531
59.928
60.325
60.721
61.118
61.515
61.912
62.309
62.706
63.103
63.5
63.896
64.293
64.690
65.087
65.484
65.881
66.278
66.675
67.071
67.468
67.865
68.262
68.659
69.056
69.453
69.85
70.246
70.643
71.040
71.437
71.834
72.231
72.628
73.025
73.421
73.818
74.215
74.612
75.009
75.406
75.803
76.2
Aviation Applications
Ratios have widespread application in the field of aviation. For example:
Compression ratio on a reciprocating engine is the
ratio of the volume of a cylinder with the piston at the
bottom of its stroke to the volume of the cylinder with
the piston at the top of its stroke. For example, a typical
compression ratio might be 10:1 (or 10 to 1).
Aspect ratio is the ratio of the length (or span) of an
airfoil to its width (or chord). A typical aspect ratio for
a commercial airliner might be 7:1 (or 7 to 1).
Air-fuel ratio is the ratio of the weight of the air to the
weight of fuel in the mixture being fed into the cylinders of a reciprocating engine. For example, a typical
air-fuel ratio might be 14.3:1 (or 14.3 to 1).
Glide ratio is the ratio of the forward distance traveled
to the vertical distance descended when an aircraft is
operating without power. For example, if an aircraft
descends 1,000 feet while it travels through the air for
a distance of two linear miles (10,560 feet), it has a
glide ratio of 10,560:1,000 which can be reduced to
10.56: 1 (or 10.56 to 1).
Gear Ratio is the number of teeth each gear represents
when two gears are used in an aircraft component. In
Figure 1-7, the pinion gear has 8 teeth and a spur gear
has 28 teeth. The gear ratio is 8:28 or 2:7.
Speed Ratio. When two gears are used in an aircraft
component, the rotational speed of each gear is represented as a speed ratio. As the number of teeth in a gear
decreases, the rotational speed of that gear increases,
and vice-versa. Therefore, the speed ratio of two gears
is the inverse (or opposite) of the gear ratio. If two gears
have a gear ratio of 2:9, then their speed ratio is 9:2.
Example: A pinion gear with 10 teeth is driving a spur
gear with 40 teeth. The spur gear is rotating at 160 rpm.
Determine the speed of the pinion gear.
Teeth in Pinion Gear
Speed of Spur Gear
=
Teeth in Spur Gear
Speed of Pinion Gear
10 teeth
160 rpm
=
40 teeth
SP (speed of pinion gear)
To solve for SP , multiply 40 × 160, then divide by 10.
The speed of the pinion gear is 640 rpm.
Example: If the cruising speed of an airplane is 200
knots and its maximum speed is 250 knots, what is
the ratio of cruising speed to maximum speed? First,
Figure 1-7. Gear ratio.
express the cruising speed as the numerator of a fraction whose denominator is the maximum speed.
Ratio =
200
250
Next, reduce the resulting fraction to its lowest
terms.
200
4
Ratio =
=
250
5
Therefore, the ratio of cruising speed to maximum
speed is 4:5.
Another common use of ratios is to convert any given
ratio to an equivalent ratio with a denominator of 1.
Example: Express the ratio 9:5 as a ratio with a denominator of 1.
R = 9 = ?
5
1
Since 9 ÷ 5 = 1.8, then 9 = 1.8
5
1
Therefore, 9:5 is the same ratio as 1.8:1. In other words,
9 to 5 is the same ratio as 1.8 to 1.
Proportion
A proportion is a statement of equality between two or
more ratios. For example,
3 = 6 or 3:4 = 6:8
4
8
This proportion is read as, “3 is to 4 as 6 is to 8.”
Extremes and Means
The first and last terms of the proportion (the 3 and 8 in
this example) are called the extremes. The second and
third terms (the 4 and 6 in this example) are called the
means. In any proportion, the product of the extremes
is equal to the product of the means.
In the proportion 2:3 = 4:6, the product of the extremes,
2 × 6, is 12; the product of the means, 3 × 4, is also
12. An inspection of any proportion will show this to
be true.
1-9
Solving Proportions
Normally when solving a proportion, three quantities
will be known, and the fourth will be unknown. To
solve for the unknown, multiply the two numbers along
the diagonal and then divide by the third number.
For example: Express the following percentages as
decimal numbers:
90%
50%
5%
150%
Example: Solve for X in the proportion given below.
65 = X
80
100
First, multiply 65 × 100: 65 × 100 = 6500
Next, divide by 80: 6500 ÷ 80 = 81.25
Therefore, X = 81.25.
=
=
=
=
.90
.50
.05
1.5
Expressing a Fraction as a Percentage
To express a fraction as a percentage, first change the
fraction to a decimal number (by dividing the numerator by the denominator), and then convert the decimal
number to a percentage as shown earlier.
Example: An airplane flying a distance of 300 miles
used 24 gallons of gasoline. How many gallons will it
need to travel 750 miles?
Example: Express the fraction 5⁄8 as a percentage.
The ratio here is: “miles to gallons;” therefore, the
proportion is set up as:
Finding a Percentage of a Given Number
This is the most common type of percentage calculation. Here are two methods to solve percentage problems: using algebra or using proportions. Each method
is shown below to find a percent of a given number.
Miles
Gallons
300 = 750
24
G
Solve for G: (750 × 24) ÷ 300 = 60
Therefore, to fly 750 miles, 60 gallons of gasoline will
be required.
Percentage
Percentage means “parts out of one hundred.” The
percentage sign is “%”. Ninety percent is expressed as
90% (= 90 parts out of 100). The decimal 0.90 equals
90⁄ , or 90 out of 100, or 90%.
100
Expressing a Decimal Number as a Percentage
To express a decimal number in percent, move the
decimal point two places to the right (adding zeros if
necessary) and then affix the percent symbol.
Example: Express the following decimal numbers as
a percent:
.90
.5
1.25
.335
=
=
=
=
90%
50%
125%
33.5%
Expressing a Percentage as a Decimal Number
Sometimes it may be necessary to express a percentage as a decimal number. To express a percentage as
a decimal number, move the decimal point two places
to the left and drop the % symbol.
1-10
5
8 = 5 ÷ 8 = 0.625 = 62.5%
Example: In a shipment of 80 wingtip lights, 15% of
the lights were defective. How many of the lights were
defective?
Algebra Method:
15% of 80 lights = N (number of defective lights)
0.15 × 80 = N
12 = N
Therefore, 12 defective lights were in the shipment.
Proportion Method:
N
15
80 = 100
To solve for N:
N × 100 = 80 × 15
N × 100 = 1200
N = 1200 ÷ 100
N = 12
or
N = (80 × 15) ÷ 100
N = 12
Finding What Percentage One Number Is
of Another
Example: A small engine rated at 12 horsepower is
found to be delivering only 10.75 horsepower. What
is the motor efficiency expressed as a percent?
Algebra Method:
N% of 12 rated horsepower = 10.75 actual horsepower
N% × 12 = 10.75
N% = 10.75 ÷ 12
N% = .8958
N = 89.58
Therefore, the motor efficiency is 89.58%.
Proportion Method:
10.75
N
12 = 100
To solve for N: N × 12 = 10.75 × 100
N × 12 = 1075
N = 1075 ÷ 12
N = 89.58
or
N = (1075 × 100) ÷ 12
N = 89.58
Therefore, the motor efficiency is 89.58%.
Finding a Number When a Percentage of It
Is Known
Example: Eighty ohms represents 52% of a microphone’s total resistance. Find the total resistance of
this microphone.
Algebraic Method:
52% of N = 80 ohms
52% × N = 80
N = 80 ÷ .52
N = 153.846
The total resistance of the microphone is 153.846
ohms.
Proportion Method:
80
52
N = 100
Solve for N: N × 52 = 80 × 100
N × 52 = 8,000
N = 8,000 ÷ 52
N = 153.846 ohms
or
N = (80 × 100) ÷ 52
N = 153.846 ohms
Positive and Negative Numbers
(Signed Numbers)
Positive numbers are numbers that are greater than
zero. Negative numbers are numbers less than zero.
[Figure 1-8] Signed numbers are also called integers.
Figure 1-8. A scale of signed numbers.
Addition of Positive and Negative Numbers
The sum (addition) of two positive numbers is positive.
The sum (addition) of two negative numbers is negative. The sum of a positive and a negative number can
be positive or negative, depending on the values of the
numbers. A good way to visualize a negative number
is to think in terms of debt. If you are in debt by $100
(or, −100) and you add $45 to your account, you are
now only $55 in debt (or −55).
Therefore: −100 + 45 = −55.
Example: The weight of an aircraft is 2,000 pounds. A
radio rack weighing 3 pounds and a transceiver weighing 10 pounds are removed from the aircraft. What is
the new weight? For weight and balance purposes, all
weight removed from an aircraft is given a minus sign,
and all weight added is given a plus sign.
2,000 + −3 + −10 = 2,000 + −13 = 1987
Therefore, the new weight is 1,987 pounds.
Subtraction of Positive and Negative Numbers
To subtract positive and negative numbers, first change
the “–” (subtraction symbol) to a “+” (addition symbol), and change the sign of the second number to
its opposite (that is, change a positive number to a
negative number or vice versa). Finally, add the two
numbers together.
Example: The daytime temperature in the city of Denver was 6° below zero (−6°). An airplane is cruising at
15,000 feet above Denver. The temperature at 15,000
feet is 20° colder than in the city of Denver. What is
the temperature at 15,000 feet?
Subtract 20 from −6: −6 – 20 = −6 + −20 = −26
The temperature is −26°, or 26° below zero at 15,000
feet above the city.
Multiplication of Positive and Negative Numbers
The product of two positive numbers is always positive.
The product of two negative numbers is always positive. The product of a positive and a negative number
is always negative.
1-11
Examples:
3 × 6 = 18 −3 × 6 = −18 −3 × −6 = 18 3 × −6 = −18
Powers
of Ten
Division of Positive and Negative Numbers
The quotient of two positive numbers is always positive. The quotient of two negative numbers is always
positive. The quotient of a positive and negative number is always negative.
Positive
Exponents
Examples:
6 ÷ 3 = 2 −6 ÷ 3 = −2 −6 ÷ −3 = 2 6 ÷ −3 = −2
Powers
The power (or exponent) of a number is a shorthand
method of indicating how many times a number, called
the base, is multiplied by itself. For example, 34 means
“3 to the power of 4.” That is, 3 multiplied by itself 4
times. The 3 is the base and 4 is the power.
Examples:
23 = 2 × 2 × 2 = 8.
Read “two to the third power equals 8.”
105 = 10 × 10 × 10 × 10 × 10 = 100,000
Read “ten to the fifth power equals 100,000.”
Special Powers
Squared. When a number has a power of 2, it is commonly referred to as “squared.” For example, 72 is read
as “seven squared” or “seven to the second power.”
To remember this, think about how a square has two
dimensions: length and width.
Cubed. When a number has a power of 3, it is commonly referred to as “cubed.” For example, 73 is
read as “seven cubed” or “seven to the third power.”
To remember this, think about how a cube has three
dimensions: length, width, and depth.
Power of Zero. Any non-zero number raised to the zero
power always equals 1.
Example:
70 = 1 1810 = 1 (-24)0 = 1
Negative Powers
A number with a negative power equals its reciprocal
with the same power made positive.
Example: The number 2-3 is read as “2 to the negative
3rd power,” and is calculated by:
1
1
1
2-3 = 23 = 2 × 2 × 2 = 8
1-12
Expansion
Value
106
10 × 10 × 10 × 10 × 10 × 10
1,000,000
105
10 × 10 × 10 × 10 × 10
100,000
104
10 × 10 × 10 × 10
10,000
103
10 × 10 × 10
1,000
102
10 × 10
100
101
10
10
100
1
Negative
Exponents
10-1
1⁄10
1⁄10 = 0.1
10-2
1⁄(10 × 10)
1⁄100 = 0.01
10-3
1⁄(10 × 10 × 10)
1⁄1,000 = 0.001
10-4
1⁄(10 × 10 × 10 × 10)
1⁄10,000 = 0.0001
10-5
1⁄(10 × 10 × 10 × 10 × 10)
1⁄100,000 = 0.00001
10-6
1⁄(10 × 10 × 10 × 10 × 10 × 10)
1⁄1,000,000 = 0.000001
Figure 1-9. Powers of ten.
When using a calculator to raise a negative number to
a power, always place parentheses around the negative
number (before raising it to a power) so that the entire
number gets raised to the power.
Law of Exponents
When multiplying numbers with powers, the powers
can be added as long as the bases are the same.
Example:
32 × 34 = (3 × 3) × (3 × 3 × 3 × 3) = 3 × 3 × 3 × 3 × 3 × 3 = 36
or 32 × 34 = 3(2+4) = 36
When dividing numbers with powers, the powers can
be subtracted as long as the bases are the same.
Example:
104 ÷ 102 =
10 × 10 × 10 × 10 10 × 10 × 10 × 10
=
= 10 × 10 = 102
10 × 10
10 × 10
or 104 ÷ 102 = 10(4 − 2) = 102
Powers of Ten
Because we use the decimal system of numbers, powers of ten are frequently seen in everyday applications.
For example, scientific notation uses powers of ten.
Also, many aircraft drawings are scaled to powers of
ten. Figure 1-9 gives more information on the powers
of ten and their values.
Roots
A root is a number that when multiplied by itself a
specified number of times will produce a given number.
The two most common roots are the square root and the
cube root. For more examples of roots, see the chart in
Figure 1-10, Functions of Numbers (on page 1-14).
Square Roots
The square root of 25, written as √25, equals 5.
That is, when the number 5 is squared (multiplied
by itself ), it produces the number 25. The symbol
√ is called a radical sign. Finding the square root of a
number is the most common application of roots. The
collection of numbers whose square roots are whole
numbers are called perfect squares. The first ten perfect
squares are: 1, 4, 9, 16, 25, 36, 49, 64, 81, and 100.
The square root of each of these numbers is 1, 2, 3, 4,
5, 6, 7, 8, 9, and 10, respectively.
For example, √36 = 6 and √81 = 9
To find the square root of a number that is not a perfect
square, use either a calculator or the estimation method.
A longhand method does exist for finding square roots,
but with the advent of calculators and because of its
lengthy explanation, it is no longer included in this
handbook. The estimation method uses the knowledge
of perfect squares to approximate the square root of
a number.
Example: Find the square root of 31. Since 31 falls
between the two perfect roots 25 and 36, we know
that √31 must be between √25 and √36. Therefore,√31
must be greater than 5 and less than 6 because √25 =
5 and √36 = 6. If you estimate the square root of 31
at 5.5, you are close to the correct answer. The square
root of 31 is actually 5.568.
Cube Roots
3
The cube root of 125, written as √125, equals 5. That
is, when the number 5 is cubed (5 multiplied by itself
then multiplying the product (25) by 5 again), it produces the number 125. It is common to confuse the
“cube” of a number with the “cube root” of a number.
For clarification, the cube of 27 = 273 = 27 × 27 × 27
3
= 19,683. However, the cube root of 27 = √27 = 3.
Fractional Powers
Another way to write a root is to use a fraction as the
power (or exponent) instead of the radical sign. The
square root of a number is written with a 1⁄2 as the
exponent instead of a radical sign. The cube root of a
number is written with an exponent of 1⁄3 and the fourth
root with an exponent of 1⁄4 and so on.
1
3
Example: √31 = 31 ⁄2 √125 = 125
1⁄
3
4
1
√16 = 16 ⁄4
Functions of Numbers Chart
The Functions of Numbers chart [Figure 1-10] is
included in this chapter for convenience in making
computations. Each column in the chart is listed below,
with new concepts explained.
• Number, (N)
• N squared, (N2)
• N cubed, (N3)
• Square root of N, (√N)
3
• Cube root of N, (√N )
• Circumference of a circle with diameter = N.
Circumference is the linear measurement of the
distance around a circle. The circumference is
calculated by multiplying the diameter of the
circle by 3.1416 (3.1416 is the number referred
to as pi, which has the symbol π). If the diameter
of a circle is 10 inches, then the circumference
would be 31.416 inches because 10 × 3.1416 =
31.4160.
• Area of a circle with diameter = N. Area of a circle
is the number of square units of measurement
contained in the circle with a diameter of N. The
area of a circle equals π multiplied by the radius
squared. This is calculated by the formula: A = π
× r2. Remember that the radius is equal to one-half
of the diameter.
Example: A cockpit instrument gauge has a round
face that is 3 inches in diameter. What is the area
of the face of the gauge? From Figure 1-10 for N
= 3, the answer is 7.0686 square inches. This is
calculated by:
If the diameter of the gauge is 3 inches, then the
radius = D⁄2 = 3⁄2 = 1.5 inches.
Area = π × r2 = 3.1416 × 1.52 = 3.1416 × 2.25 =
7.0686 square inches.
Scientific Notation
Scientific notation is used as a type of shorthand to
express very large or very small numbers. It is a way
to write numbers so that they do not take up as much
space on the page. The format of a number written
in scientific notation has two parts. The first part is a
number greater than or equal to 1 and less than 10 (for
example, 2.35). The second part is a power of 10 (for
example, 106). The number 2,350,000 is expressed in
scientific notation as 2.35 × 106. It is important that
the decimal point is always placed to the right of the
first digit. Notice that very large numbers always have
1-13
Number
Square
Cube
Square Root
Cube Root
Circumference
Area
Number (N)
N Squared (N2)
N Cubed (N3)
Square Root
of N (√¯¯N̄ )
Cube Root
of N ( √3¯¯N̄ )
Circumference of a circle
with diameter = N
Area of a circle
with diameter = N
1
1
1
1.000
1.000
3.142
0.785
2
4
8
1.414
1.260
6.283
3.142
3
9
27
1.732
1.442
9.425
7.069
4
16
64
2.000
1.587
12.566
12.566
5
25
125
2.236
1.710
15.708
19.635
6
36
216
2.449
1.817
18.850
28.274
7
49
343
2.646
1.913
21.991
38.484
8
64
512
2.828
2.000
25.133
50.265
9
81
729
3.000
2.080
28.274
63.617
10
100
1,000
3.162
2.154
31.416
78.540
11
121
1,331
3.317
2.224
34.558
95.033
12
144
1,728
3.464
2.289
37.699
113.01
13
169
2,197
3.606
2.351
40.841
132.73
14
196
2,744
3.742
2.410
43.982
153.94
15
225
3,375
3.873
2.466
47.124
176.71
16
256
4,096
4.000
2.520
50.265
201.06
17
289
4,913
4.123
2.571
53.407
226.98
18
324
5,832
4.243
2.621
56.549
254.47
19
361
6,859
4.359
2.668
59.690
283.53
20
400
8,000
4.472
2.714
62.832
314.16
21
441
9,261
4.583
2.759
65.973
346.36
22
484
10,648
4.690
2.802
69.115
380.13
23
529
12,167
4.796
2.844
72.257
415.48
24
576
13,824
4.899
2.885
75.398
452.39
25
625
15,625
5.000
2.924
78.540
490.87
26
676
17,576
5.099
2.963
81.681
530.93
27
729
19,683
5.196
3.000
84.823
572.55
28
784
21,952
5.292
3.037
87.965
615.75
29
841
24,389
5.385
3.072
91.106
660.52
30
900
27,000
5.477
3.107
94.248
706.86
31
961
29,791
5.568
3.141
97.389
754.77
32
1,024
32,768
5.657
3.175
100.531
804.25
33
1,089
35,937
5.745
3.208
103.672
855.30
34
1,156
39,304
5.831
3.240
106.814
907.92
35
1,225
42,875
5.916
3.271
109.956
962.11
36
1,296
46,656
6.000
3.302
113.097
1017.88
37
1,369
50,653
6.083
3.332
116.239
1075.21
38
1,444
54,872
6.164
3.362
119.380
1134.11
39
1,521
59,319
6.245
3.391
122.522
1194.59
40
1,600
64,000
6.325
3.420
125.664
1256.64
41
1,681
68,921
6.403
3.448
128.805
1320.25
42
1,764
74,088
6.481
3.476
131.947
1385.44
43
1,849
79,507
6.557
3.503
135.088
1452.20
44
1,936
85,184
6.633
3.530
138.230
1520.53
45
2,025
91,125
6.708
3.557
141.372
1590.43
46
2,116
97,336
6.782
3.583
144.513
1661.90
47
2,209
103,823
6.856
3.609
147.655
1734.94
48
2,304
110,592
6.928
3.634
150.796
1809.56
49
2,401
117,649
7.000
3.659
153.938
1885.74
50
2,500
125,000
7.071
3.684
157.080
1963.49
Figure 1-10. Functions of numbers.
1-14
Number
Square
Cube
Square Root
Cube Root
Circumference
Area
Number (N)
N Squared (N2)
N Cubed (N3)
Square Root
of N (√¯¯N̄ )
Cube Root
of N ( √3¯¯N̄ )
Circumference of a circle
with diameter = N
Area of a circle
with diameter = N
51
2,601
132,651
7.141
3.708
160.221
2042.82
52
2,704
140,608
7.211
3.733
163.363
2123.71
53
2,809
148,877
7.280
3.756
166.504
2206.18
54
2,916
157,464
7.348
3.780
169.646
2290.22
55
3,025
166,375
7.416
3.803
172.787
2375.83
56
3,136
175,616
7.483
3.826
175.929
2463.01
57
3,249
185,193
7.550
3.849
179.071
2551.76
58
3,364
195,112
7.616
3.871
182.212
2642.08
59
3,481
205,379
7.681
3.893
185.354
2733.97
60
3,600
216,000
7.746
3.915
188.495
2827.43
61
3,721
226,981
7.810
3.937
191.637
2922.46
62
3,844
238,328
7.874
3.958
194.779
3019.07
63
3,969
250,047
7.937
3.979
197.920
3117.24
64
4,096
262,144
8.000
4.000
201.062
3216.99
65
4,225
274,625
8.062
4.021
204.203
3318.30
66
4,356
287,496
8.124
4.041
207.345
3421.19
67
4,489
300,763
8.185
4.062
210.487
3525.65
68
4,624
314,432
8.246
4.082
213.628
3631.68
69
4,761
328,509
8.307
4.102
216.770
3739.28
70
4,900
343,000
8.367
4.121
219.911
3848.45
71
5,041
357,911
8.426
4.141
223.053
3959.19
72
5,184
373,248
8.485
4.160
226.194
4071.50
73
5,329
389,017
8.544
4.179
229.336
4185.38
74
5,476
405,224
8.602
4.198
232.478
4300.84
75
5,625
421,875
8.660
4.217
235.619
4417.86
76
5,776
438,976
8.718
4.236
238.761
4536.46
77
5,929
456,533
8.775
4.254
241.902
4656.62
78
6,084
474,552
8.832
4.273
245.044
4778.36
79
6,241
493,039
8.888
4.291
248.186
4901.67
80
6,400
512,000
8.944
4.309
251.327
5026.54
81
6,561
531,441
9.000
4.327
254.469
5152.99
82
6,724
551,368
9.055
4.344
257.610
5281.01
83
6,889
571,787
9.110
4.362
260.752
5410.60
84
7,056
592,704
9.165
4.380
263.894
5541.76
85
7,225
614,125
9.220
4.397
267.035
5674.50
86
7,396
636,056
9.274
4.414
270.177
5808.80
87
7,569
658,503
9.327
4.431
273.318
5944.67
88
7,744
681,472
9.381
4.448
276.460
6082.12
89
7,921
704,969
9.434
4.465
279.602
6221.13
90
8,100
729,000
9.487
4.481
282.743
6361.72
91
8,281
753,571
9.539
4.498
285.885
6503.88
92
8,464
778,688
9.592
4.514
289.026
6647.60
93
8,649
804,357
9.644
4.531
292.168
6792.90
94
8,836
830,584
9.695
4.547
295.309
6939.77
95
9,025
857,375
9.747
4.563
298.451
7088.21
96
9,216
884,736
9.798
4.579
301.593
7238.22
97
9,409
912,673
9.849
4.595
304.734
7389.81
98
9,604
941,192
9.900
4.610
307.876
7542.96
99
9,801
970,299
9.950
4.626
311.017
7697.68
100
10,000
1,000,000
10.000
4.642
314.159
7853.98
Figure 1-10. Functions of numbers. (continued)
1-15
a positive power of 10 and very small numbers always
have a negative power of 10.
Example: The velocity of the speed of light is over
186,000,000 mph. This can be expressed as 1.86 × 108
mph in scientific notation. The mass of an electron is
approximately 0.000,000,000,000,000,000,000,000
,000,911 grams. This can be expressed in scientific
notation as 9.11 × 10-28 grams.
Converting Numbers from Standard Notation to
Scientific Notation
Example: Convert 1,244,000,000,000 to scientific
notation as follows. First, note that the decimal point
is to the right of the last zero. (Even though it is not
usually written, it is assumed to be there.)
Conversion
Large numbers
with positive
powers of 10
Small numbers
with negative
powers of 10
From standard
notation to scientific
notation
Move decimal place
to the left
Move decimal place
to the right
From scientific
notation to standard
notation
Move decimal place
to the right
Move decimal place
to the left
Figure 1-11. Converting between scientific and
standard notation.
1,244,000,000,000 = 1,244,000,000,000
When converting, remember that large numbers always
have positive powers of ten and small numbers always
have negative powers of ten. Refer to Figure 1-11
to determine which direction to move the decimal
point.
To change to the format of scientific notation, the decimal point must be moved to the position between the
first and second digits, which in this case is between
the 1 and the 2. Since the decimal point must be moved
12 places to the left to get there, the power of 10 will
be 12. Remember that large numbers always have a
positive exponent. Therefore, 1,244,000,000,000 =
1.244 × 1012 when written in scientific notation.
Addition, Subtraction, Multiplication, and
Division of Scientific Numbers
To add, subtract, multiply, or divide numbers in scientific notation, change the scientific notation number
back to standard notation. Then add, subtract, multiply
or divide the standard notation numbers. After the
computation, change the final standard notation number
back to scientific notation.
Example: Convert 0.000000457 from standard notation to scientific notation. To change to the format of
scientific notation, the decimal point must be moved
to the position between the first and second numbers,
which in this case is between the 4 and the 5. Since
the decimal point must be moved 7 places to the right
to get there, the power of 10 will be −7. Remember
that small numbers (those less than one) will have a
negative exponent. Therefore, 0.000000457 = 4.57 ×
10-7 when written in scientific notation.
Converting Numbers from Scientific Notation to
Standard Notation
Example: Convert 3.68 × 107 from scientific notation
to standard notation, as follows. To convert from scientific notation to standard notation, move the decimal
place 7 places to the right. 3.68 × 107 = 36800000 =
36,800,000. Another way to think about the conversion
is 3.68 × 107 = 3.68 × 10,000,000 = 36,800,000.
Example: Convert 7.1543 × 10-10 from scientific notation to standard notation. Move the decimal place 10
places to the left: 7.1543 × 10-10 =.00000000071543.
Another way to think about the conversion is 7.1543 ×
10-10 = 7.1543 × .0000000001 = .00000000071543
1-16
Algebra
Algebra is the branch of mathematics that uses letters
or symbols to represent variables in formulas and
equations.
For example, in the equation D = V × T, where Distance
= Velocity × Time, the variables are: D, V, and T.
Equations
Algebraic equations are frequently used in aviation to
show the relationship between two or more variables.
Equations normally have an equals sign (=) in the
expression.
Example: The formula A = π × r2 shows the relationship
between the area of a circle (A) and the length of the
radius (r) of the circle. The area of a circle is equal to π
(3.1416) times the radius squared. Therefore, the larger
the radius, the larger the area of the circle.
Algebraic Rules
When solving for a variable in an equation, you can
add, subtract, multiply or divide the terms in the
equation, you do the same to both sides of the equals
sign.
Examples: Solve the following equations for the
value N.
3N = 21
To solve for N, divide both sides by 3.
3N ÷ 3 = 21 ÷ 3
N=7
N + 17 = 59
To solve for N, subtract 17 from both sides.
N + 17 – 17 = 59 – 17
N = 42
N – 22 = 100
To solve for N, add 22 to both sides.
N – 22 + 22 = 100 + 22
N = 122
N
5 = 50
To solve for N, multiply both sides by 5.
N
5 × 5 = 50 × 5
N = 250
Solving for a Variable
Another application of algebra is to solve an equation
for a given variable.
Example: Using the formula given in Figure 1-12, find
the total capacitance (CT) of the series circuit containing three capacitors with
C1 = .1 microfarad
C2 = .015 microfarad
C3 = .05 microfarad
First, substitute the given values into the formula:
Figure 1-12. Total capacitance in a series circuit.
1
1
1
CT =
=
=
1
1
1
1
1
1
10 + 66.66 + 20
+ +
+
+
C1 C2 C3 0.1 0.015 0.05
Therefore, CT = 1⁄96.66 = .01034 microfarad. The
microfarad (10-6 farad) is a unit of measurement of
capacitance. This will be discussed in greater length
beginning on page 10-51 in chapter 10, Electricity.
Use of Parentheses
In algebraic equations, parentheses are used to group
numbers or symbols together. The use of parentheses
helps us to identify the order in which we should apply
mathematical operations. The operations inside the
parentheses are always performed first in algebraic
equations.
Example: Solve the algebraic equation N = (4 + 3) 2.
First, perform the operation inside the parentheses.
That is, 4 + 3 = 7. Then complete the exponent calculation N = (7) 2 = 7 × 7 = 49.
When using more complex equations, which may combine several terms and use multiple operations, grouping the terms together helps organize the equation.
Parentheses, ( ), are most commonly used in grouping,
but you may also see brackets, [ ]. When a term or
expression is inside one of these grouping symbols, it
means that any operation indicated to be done on the
group is done to the entire term or expression.
Example:
Solve the equation N = 2 × [(9 ÷ 3) + (4 + 3)2]. Start
with the operations inside the parentheses ( ), then
perform the operations inside the brackets [ ].
N = 2 × [(9 ÷ 3) + (4 + 3)2]
N = 2 × [3 + (7)2] First, complete the operations
inside the parentheses ( ).
N = 2 × [3 + 49]
N = 2 × [52] Second, complete the operations inside
the brackets [ ].
N = 104
Order of Operation
In algebra, rules have been set for the order in which
operations are evaluated. These same universally
accepted rules are also used when programming
algebraic equations in calculators. When solving the
following equation, the order of operation is given
below:
N = (62 – 54)2 + 62 – 4 + 3 × [8 + (10 ÷ 2)] + √25 + (42 × 2) ÷ 4 + 3⁄4
1. Parentheses. First, do everything in parentheses,
( ). Starting from the innermost parentheses. If the
expression has a set of brackets, [ ], treat these
exactly like parentheses. If you are working with
a fraction, treat the top as if it were in parentheses
and the denominator as if it were in parentheses,
even if there are none shown. From the equation
1-17
above, completing the calculation in parentheses
gives the following:
N = (8)2 + 62 – 4 + 3 × [8 + (5)] + √25 + (84) ÷ 4 + 3⁄4,
then
N = (8)2 + 62 – 4 + 3 × [13] + √25 + 84 ÷ 4 + 3⁄4
2. Exponents. Next, clear any exponents. Treat any
roots (square roots, cube roots, and so forth) as
exponents. Completing the exponents and roots
in the equation gives the following:
N = 64 + 36 – 4 + 3 × 13 + 5 + 84 ÷ 4 + 3⁄4
3. Multiplication and Division. Evaluate all of the
multiplications and divisions from left to right.
Multiply and divide from left to right in one step.
A common error is to use two steps for this (that is,
to clear all of the multiplication signs and then clear
all of the division signs), but this is not the correct
method. Treat fractions as division. Completing the
multiplication and division in the equation gives
the following:
N = 64 + 36 – 4 + 39 + 5 + 21 + 3⁄4
4. Addition and Subtraction. Evaluate the additions
and subtractions from left to right. Like above,
addition and subtraction are computed left to
right in one step. Completing the addition and
subtraction in the equation gives the following:
X = 161 3⁄4
Computing Area of
Two-dimensional Solids
Area is a measurement of the amount of surface of an
object. Area is usually expressed in such units as square
inches or square centimeters for small surfaces or in
square feet or square meters for larger surfaces.
Rectangle
A rectangle is a four-sided figure with opposite sides
of equal length and parallel. [Figure 1-13] All of the
angles are right angles. A right angle is a 90° angle.
The rectangle is a very familiar shape in mechanics.
The formula for the area of a rectangle is:
Area = Length × Width = L × W
Example: An aircraft floor panel is in the form of a
rectangle having a length of 24 inches and a width of
12 inches. What is the area of the panel expressed in
square inches? First, determine the known values and
substitute them in the formula.
A = L × W = 24 inches × 12 inches = 288 square inches
Square
A square is a four-sided figure with all sides of equal
length and parallel. [Figure 1-14] All angles are right
angles. The formula for the area of a square is:
Area = Length × Width = L × W
Order of Operation for Algebraic Equations
1. Parentheses
2. Exponents
3. Multiplication and Division
4. Addition and Subtraction
Use the acronym PEMDAS to remember the order
of operation in algebra. PEMDAS is an acronym
for parentheses, exponents, multiplication, division,
addition, and subtraction. To remember it, many use
the sentence, “Please Excuse My Dear Aunt Sally.”
Always remember, however, to multiply/divide or
add/subtract in one sweep from left to right, not
separately.
Figure 1-13. Rectangle.
Figure 1-14. Square.
1-18
Since the length and the width of a square are the same
value, the formula for the area of a square can also be
written as:
Area = Side × Side = S2
Example: What is the area of a square access plate
whose side measures 25 inches? First, determine the
known value and substitute it in the formula.
Figure 1-15. Types of triangles.
A = L × W = 25 inches × 25 inches = 625 square inches
Triangle
A triangle is a three-sided figure. The sum of the three
angles in a triangle is always equal to 180°. Triangles
are often classified by their sides. An equilateral triangle has 3 sides of equal length. An isosceles triangle
has 2 sides of equal length. A scalene triangle has three
sides of differing length. Triangles can also be classified by their angles: An acute triangle has all three
angles less than 90°. A right triangle has one right angle
(a 90° angle). An obtuse triangle has one angle greater
than 90°. Each of these types of triangles is shown in
Figure 1-15.
Figure 1-16. Obtuse triangle.
The formula for the area of a triangle is
Area = 1⁄2 × (Base × Height) = 1⁄2 × (B × H)
Example: Find the area of the obtuse triangle shown
in Figure 1-16. First, substitute the known values in
the area formula.
Figure 1-17. Parallelogram.
A = 1⁄2 × (B × H) = 1⁄2 × (2'6" × 3'2")
Next, convert all dimensions to inches:
2'6" = (2 × 12") + 6" = (24 + 6) = 30 inches
3'2" = (3 × 12") + 2" = (36 + 2) = 38 inches
Now, solve the formula for the unknown value:
A = 1⁄2 × (30 inches × 38 inches) = 570 square inches
Figure 1-18. Trapezoid.
Parallelogram
A parallelogram is a four-sided figure with two pairs
of parallel sides. [Figure 1-17] Parallelograms do not
necessarily have four right angles. The formula for the
area of a parallelogram is:
Trapezoid
A trapezoid is a four-sided figure with one pair of
parallel sides. [Figure 1-18] The formula for the area
of a trapezoid is:
Area = Length × Height = L × H
Area = 1⁄2 (Base1 + Base2) × Height
Example: What is the area of a trapezoid in Figure 1-19
whose bases are 14 inches and 10 inches, and whose
height (or altitude) is 6 inches? First, substitute the
known values in the formula.
1-19
Figure 1-19. Trapezoid, with dimensions.
A = 1⁄2 (b1 + b2) × H
= 1⁄2 (14 inches + 10 inches) × 6 inches
A = 1⁄2 (24 inches) × 6 inches
= 12 inches × 6 inches = 72 square inches.
Circle
A circle is a closed, curved, plane figure. [Figure 1‑20]
Every point on the circle is an equal distance from
the center of the circle. The diameter is the distance
across the circle (through the center). The radius is the
distance from the center to the edge of the circle. The
diameter is always twice the length of the radius. The
circumference, or distance around, a circle is equal to
the diameter times π.
Circumference = C = d π
The formula for the area of a circle is:
Area = π × radius2 = π × r2
Example: The bore, or “inside diameter,” of a certain
aircraft engine cylinder is 5 inches. Find the area of
the cross section of the cylinder.
Figure 1-21. Ellipse.
First, substitute the known values in the formula:
A = π × r2.
The diameter is 5 inches, so the radius is 2.5 inches.
(diameter = radius × 2)
A = 3.1416 × (2.5 inches)2 = 3.1416 × 6.25 square
inches = 19.635 square inches
Ellipse
An ellipse is a closed, curved, plane figure and is commonly called an oval. [Figure 1-21] In a radial engine,
the articulating rods connect to the hub by pins, which
travel in the pattern of an ellipse (i.e., an elliptical or
obital path).
Wing Area
To describe the shape of a wing [Figure 1-23], several
terms are required. To calculate wing area, it will be
necessary to know the meaning of the terms “span”
and “chord.” The wingspan, S, is the length of the wing
from wingtip to wingtip. The chord is the average width
Figure 1-20. Circle.
1-20
Figure 1-22. Wing planform.
Object
Area
Formula
Figure
Rectangle
Length × Width
A=L×W
1-13
Square
Length × Width or
Side × Side
A = L × W or A = S2
1-14
Triangle
½ × (Length × Height) or
½ × (Base × Height) or
(Base × Height) ÷ 2
A = ½ (L × H) or
A = ½ (B × H) or
A = (B × H) ÷ 2
1-15
Parallelogram
Length × Height
A=L×H
1-17
Trapezoid
½ (base 1 + base 2 ) × Height
A = ½ (b1 + b2) × H
1-18
Circle
π × radius 2
A = π × r2
1-20
Ellipse
π × semi-axis A × semi-axis B
A=π×A×B
1-21
Figure 1-23. Formulas to compute area.
of the wing from leading edge to trailing edge. If the
wing is a tapered wing, the average width, known as
the mean chord (C), must be known to find the area.
The formula for calculating wing area is:
Area of a wing = Span × Mean Chord
Example: Find the area of a tapered wing whose span is
50 feet and whose mean chord is 6'8". First, substitute
the known values in the formula.
A= S × C
= 50 feet × 6 feet 8 inches
(Note: 8 inches = 8⁄12 feet = .67 feet)
= 50 feet × 6.67 feet
= 333.5 square feet
Units of Area
A square foot measures 1 foot by 1 foot. It also measures 12 inches by 12 inches. Therefore, one square
foot also equals 144 square inches (that is, 12 × 12 =
144). To convert square feet to square inches, multiply
by 144. To convert square inches to square feet, divide
by 144.
A square yard measures 1 yard by 1 yard. It also measures 3 feet by 3 feet. Therefore, one square yard also
equals 9 square feet (that is, 3 × 3 = 9). To convert
square yards to square feet, multiply by 9. To convert
square feet to square yards, divide by 9. Refer to Figure 1-37, Applied Mathematics Formula Sheet, at the
end of the chapter for a comparison of different units
of area.
Computing Volume of
Three-Dimensional Solids
Three-dimensional solids have length, width, and
height. There are many three-dimensional solids,
but the most common are rectangular solids, cubes,
cylinders, spheres, and cones. Volume is the amount
of space within a solid. Volume is expressed in cubic
units. Cubic inches or cubic centimeters are used for
small spaces and cubic feet or cubic meters for larger
spaces.
Rectangular Solid
A rectangular solid is a three-dimensional solid with six
rectangle-shaped sides. [Figure 1-24] The volume is the
number of cubic units within the rectangular solid. The
formula for the volume of a rectangular solid is:
Volume = Length × Width × Height = L × W × H
In Figure 1-24, the rectangular solid is 3 feet by 2 feet
by 2 feet.
The volume of the solid in Figure 1-24 is = 3 ft × 2 ft
× 2 ft = 12 cubic feet.
Figure 1-23 summarizes the formulas for computing
the area of two-dimensional solids.
Figure 1-24. Rectangular solid.
1-21
Example: A rectangular baggage compartment measures 5 feet 6 inches in length, 3 feet 4 inches in width,
and 2 feet 3 inches in height. How many cubic feet of
baggage will it hold? First, substitute the known values
into the formula.
V= L×W× H
= 5'6" × 3'4" × 2'3"
= 5.5 ft × 3.33 ft × 2.25 ft
= 41.25 cubic feet
Cube
A cube is a solid with six square sides. [Figure 1-25] A
cube is just a special type of rectangular solid. It has the
same formula for volume as does the rectangular solid
which is Volume = Length × Width × Height = L × W
× H. Because all of the sides of a cube are equal, the
volume formula for a cube can also be written as:
Volume = Side × Side × Side = S3
Example: A large, cube-shaped carton contains a
shipment of smaller boxes inside of it. Each of the
smaller boxes is 1 ft × 1 ft × 1 ft. The measurement of
the large carton is 3 ft × 3 ft × 3 ft. How many of the
smaller boxes are in the large carton? First, substitute
the known values into the formula.
V= L×W× H
= 3 ft × 3 ft × 3 ft
= 27 cubic feet of volume in the large carton
Since each of the smaller boxes has a volume of 1 cubic
foot, the large carton will hold 27 boxes.
Figure 1-26. Cylinder.
Cylinder
A solid having the shape of a can, or a length of pipe,
or a barrel is called a cylinder. [Figure 1-26] The ends
of a cylinder are identical circles. The formula for the
volume of a cylinder is:
Volume = π × radius2 × height of the cylinder = π r2 × H
One of the most important applications of the volume
of a cylinder is finding the piston displacement of a
cylinder in a reciprocating engine. Piston displacement
is the total volume (in cubic inches, cubic centimeters,
or liters) swept by all of the pistons of a reciprocating
engine as they move in one revolution of the crankshaft.
The formula for piston displacement is given as:
Piston Displacement =
π × (bore divided by 2)2 × stroke × (# cylinders)
The bore of an engine is the inside diameter of the cylinder. The stroke of the engine is the length the piston
travels inside the cylinder. [Figure 1-27]
Example: Find the piston displacement of one cylinder
in a multi-cylinder aircraft engine. The engine has a
cylinder bore of 5.5 inches and a stroke of 5.4 inches.
First, substitute the known values in the formula.
V = π × r2 × h = (3.1416) × (5.5 ÷ 2)2 × (5.4)
V = 23.758 × 5.4 = 128.29 cubic inches
Figure 1-25. Cube.
The piston displacement of one cylinder is 128.29
cubic inches. For an eight cylinder engine, then the
total engine displacement would be:
Total Displacement for 8 cylinders = 8 × 128.29 =
1026.32 cubic inches of displacement
1-22
Figure 1-29. Cone.
Cone
A solid with a circle as a base and with sides that gradually taper to a point is called a cone. [Figure 1-29] The
formula for the volume of a cone is given as:
Figure 1-27. Cylinder displacement.
Sphere
A solid having the shape of a ball is called a sphere.
[Figure 1-28] A sphere has a constant diameter. The
radius (r) of a sphere is one-half of the diameter (D).
The formula for the volume of a sphere is given as:
V = 4⁄3 × π × radius3 = 4⁄3 × π × r3 or V = 1⁄6 × πD3
Example: A pressure tank inside the fuselage of a cargo
aircraft is in the shape of a sphere with a diameter of
34 inches. What is the volume of the pressure tank?
V = 4⁄3 × π × radius3 = 4⁄3 × (3.1416) × (34⁄2)3
= 1.33 × 3.1416 × 173 = 1.33 × 3.1416 × 4913
V = 20,528.125 cubic inches
V = 1⁄3 × π × radius2 × height = 1⁄3 × π × r2 × H
Units of Volume
Since all volumes are not measured in the same units,
it is necessary to know all the common units of volume
and how they are related to each other. For example, the
mechanic may know the volume of a tank in cubic feet
or cubic inches, but when the tank is full of gasoline,
he or she will be interested in how many gallons it
contains. Refer to Figure 1-37, Applied Mathematics
Formula Sheet, at the end of the chapter for a comparison of different units of volume.
Computing Surface Area of
Three-dimensional Solids
The surface area of a three-dimensional solid is the
sum of the areas of the faces of the solid. Surface
area is a different concept from that of volume. For
example, surface area is the amount of sheet metal
needed to build a rectangular fuel tank while volume
is the amount of fuel that the tank can contain.
Rectangular Solid
The formula for the surface area of a rectangular solid
[Figure 1-24] is given as:
Surface Area =
2 × [(Width × Length) + (Width × Height) + (Length × Height)]
= 2 × [(W × L) + (W × H) + (L × H)]
Figure 1-28. Sphere.
1-23
Cube
The formula for the surface area of a cube [Figure
1‑25] is given as:
Surface Area = 6 × (Side × Side) = 6 × S2
Example: What is the surface area of a cube with a side
measure of 8 inches?
Surface Area = 6 × (Side × Side)
= 6 × S2 = 6 × 82 = 6 × 64
= 384 square inches
Cylinder
The formula for the surface area of a cylinder [Figure
1-26] is given as:
Surface Area = 2 × π × radius 2 + π × diameter × height
= 2 × π × r2 + π × D × H
Sphere
The formula for the surface area of a sphere [Figure
1-28] is given as:
Surface Area = 4 × π × radius2 = 4 × π × r2
Cone
The formula for the surface area of a right circular cone
[Figure 1-29] is given as:
Surface Area = π × radius × [radius + (radius2
+ height2)1⁄2]
= π × r × [r + (r2 + H2)1⁄2]
Figure 1-30 summarizes the formulas for computing
the volume and surface area of three-dimensional
solids.
Solid
Volume
Surface Area
Figure
Rectangular
Solid
L×W×H
2 × [(W × L) + (W × H) +
(L × H)]
1-23
Cube
S3
6 × S2
1-24
Cylinder
π × r2 × H
2 × π × r2 + π × D × H
1-25
Sphere
4⁄ × π × r3
3
4 × π × r2
1-27
Cone
1⁄ × π × r2 × H
3
π × r × [r + (r2 + H2)1⁄2]
1-28
Trigonometric Functions
Trigonometry is the study of the relationship between
the angles and sides of a triangle. The word trigonometry comes from the Greek trigonon, which means three
angles, and metro, which means measure.
Right Triangle, Sides and Angles
In Figure 1-31, notice that each angle is labeled with a
capital letter. Across from each angle is a corresponding
side, each labeled with a lower case letter. This triangle
is a right triangle because angle C is a 90° angle. Side
a is opposite from angle A, and is sometimes referred
to as the “opposite side.” Side b is next to, or adjacent
to, angle A and is therefore referred to as the “adjacent
side.” Side c is always across from the right angle and
is referred to as the “hypotenuse.”
Sine, Cosine, and Tangent
The three primary trigonometric functions and
their abbreviations are: sine (sin), cosine (cos),
and tangent (tan). These three functions can be
found on most scientific calculators. The three
trigonometric functions are actually ratios comparing two of the sides of the triangle as follows:
Sine (sin) of angle A =
opposite side (side a)
hypotenuse (side c)
Cosine (cos) of angle A =
adjacent side (side b)
hypotenuse (side c)
opposite side (side a)
Tangent (tan) of angle A = adjacent side (side b)
Example: Find the sine of a 30° angle.
Calculator Method:
Using a calculator, select the “sin” feature, enter the
number 30, and press “enter.” The calculator should
display the answer as 0.5. This means that when angle
Figure 1-30. Formulas to compute volume
and surface area.
Figure 1-31. Right triangle.
1-24
A equals 30°, then the ratio of the opposite side (a) to
the hypotenuse (c) equals 0.5 to 1, so the hypotenuse
is twice as long as the opposite side for a 30° angle.
Therefore, sin 30° = 0.5.
Trigonometric Table Method:
When using a trigonometry table, find 30° in the first
column. Next, find the value for sin 30° under the
second column marked “sine” or “sin.” The value for
sin 30° should be 0.5.
Pythagorean Theorem
The Pythagorean Theorem is named after the ancient
Greek mathematician, Pythagoras (~500 B.C.). This
theorem is used to find the third side of any right
triangle when two sides are known. The Pythagorean
Theorem states that a2 + b2 = c2. [Figure 1-32] Where
c = the hypotenuse of a right triangle, a is one side of
the triangle and b is the other side of the triangle.
Example: What is the length of the longest side of a
right triangle, given the other sides are 7 inches and 9
inches? The longest side of a right triangle is always
side c, the hypotenuse. Use the Pythagorean Theorem
to solve for the length of side c as follows:
a2 + b2 = c2
72 + 92 = c2
49 + 81 = c2
130 = c2
If c2 = 130 then c = √130 = 11.4 inches
Therefore, side c = 11.4 inches.
Example: The cargo door opening in a military airplane
is a rectangle that is 5 1⁄ 2 feet tall by 7 feet wide. A section of square steel plate that is 8 feet wide by 8 feet
tall by 1 inch thick must fit inside the airplane. Can
the square section of steel plate fit through the cargo
Figure 1-32. Pythagorean Theorem.
door? It is obvious that the square steel plate will not
fit horizontally through the cargo door. The steel plate
is 8 feet wide and the cargo door is only 7 feet wide.
However, if the steel plate is tilted diagonally, will it
fit through the cargo door opening?
The diagonal distance across the cargo door opening
can be calculated using the Pythagorean Theorem
where “a” is the cargo door height, “b” is the cargo
door width, and “c” is the diagonal distance across the
cargo door opening.
a2 + b2 = c2
(5.5 ft)2 + (7 ft)2 = c2
30.25 + 49 = c2
79.25 = c2 c = 8.9 ft
The diagonal distance across the cargo door opening
is 8.9 feet, so the 8-foot wide square steel plate will
fit diagonally through the cargo door opening and into
the airplane.
Measurement Systems
Conventional (U.S. or English) System
Our conventional (U.S. or English) system of measurement is part of our cultural heritage from the days
when the thirteen colonies were under British rule. It
started as a collection of Anglo-Saxon, Roman, and
Norman-French weights and measures. For example,
the inch represents the width of the thumb and the
foot is from the length of the human foot. Tradition
holds that King Henry I decreed that the yard should
be the distance from the tip of his nose to the end
of his thumb. Since medieval times, commissions
appointed by various English monarchs have reduced
the chaos of measurement by setting specific standards
for some of the most important units. Some of the
conventional units of measure are: inches, feet, yards,
miles, ounces, pints, gallons, and pounds. Because
the conventional system was not set up systematically, it contains a random collection of conversions.
For example, 1 mile = 5,280 feet and 1 foot = 12
inches.
Metric System
The metric system, also known as the International
System of Units (SI), is the dominant language of
measurement used today. Its standardization and
decimal features make it well-suited for engineering
and aviation work.
1-25
The metric system was first envisioned by Gabriel
Mouton, Vicar of St. Paul’s Church in Lyons, France.
The meter is the unit of length in the metric system,
and it is equal to one ten-millionth of the distance from
the equator to the North Pole. The liter is the unit of
volume and is equal to one cubic decimeter. The gram
is the unit of mass and is equal to one cubic centimeter
of water.
All of the metric units follow a consistent naming
scheme, which consists of attaching a prefix to the unit.
For example, since kilo stands for 1,000 one kilometer
equals 1,000 meters. Centi is the prefix for one hundredth, so one meter equals one hundred centimeters.
Milli is the prefix for one thousandths and one gram
equals one thousand milligrams. Refer to Figure 1-33
for the names and definitions of metric prefixes.
Measurement Systems and Conversions
The United States primarily uses the conventional (U.S.
or English) system, although it is slowly integrating the
metric system (SI). A recommendation to transition to
the metric system within ten years was initiated in the
1970s. However, this movement lost momentum, and
the United States continues to use both measurement
systems. Therefore, information to convert between the
conventional (U.S., or English) system and the metric
(SI) system has been included in Figure 1-37, Applied
Mathematics Formula Sheet, at the end of this chapter.
Examples of its use are as follows:
To convert inches to millimeters, multiply the number
of inches by 25.4.
Example: 20 inches = 20 × 25.4 = 508 mm
Prefix
Means
tera (1012)
One trillion times
giga (109)
One billion times
mega (106)
One million times
kilo (103)
One thousand times
hecto (102)
One hundred times
deca (101)
Ten times
deci (10-1)
One tenth of
centi (10-2)
One hundredth of
milli (10-3)
One thousandth of
micro (10-6)
One millionth of
nano (10-9)
One billionth of
pico (10-12)
One trillionth of
Figure 1-33. Names and definitions of metric prefixes.
1-26
To convert ounces to grams, multiply the number of
ounces by 28.35.
Example: 12 ounces = 12 × 28.35 = 340.2 grams
The Binary Number System
The binary number system has only two digits: 0 and 1.
The prefix in the word “binary” is a Latin root for the
word “two” and its use was first published in the late
1700s. The use of the binary number system is based
on the fact that switches or valves have two states:
open or closed (on/off).
Currently, one of the primary uses of the binary number system is in computer applications. Information
is stored as a series of 0s and 1s, forming strings of
binary numbers. An early electronic computer, ENIAC
(Electronic Numerical Integrator And Calculator), was
built in 1946 at the University of Pennsylvania and
contained 17,000 vacuum tubes, along with 70,000
resistors, 10,000 capacitors, 1,500 relays, 6,000 manual
switches and 5 million soldered joints. Computers
obviously have changed a great deal since then, but
are still based on the same binary number system. The
binary number system is also useful when working with
digital electronics because the two basic conditions of
electricity, on and off, can be represented by the two
digits of the binary number system. When the system
is on, it is represented by the digit 1, and when it is off,
it is represented by the digit zero.
Place Values
The binary number system is a base-2 system. That
is, the place values in the binary number system are
based on powers of 2. An 8-bit binary number system
is shown in Figure 1-34 on the next page.
Converting Binary Numbers to Decimal Numbers
To convert a binary number to a decimal number, add
up the place values that have a 1 (place values that
have a zero do not contribute to the decimal number
conversion).
Example: Convert the binary number 10110011 to a
decimal number. Using the Place Value chart shown in
Figure 1-35, add up the place values of the ‘1s’ in the
binary number (ignore the place values with a zero in
the binary number).
The binary number 10110011
= 128 + 0 + 32 + 16 + 0 + 0 + 2 + 1
= 179 in the decimal number system
Place Value
27
26
25
or 128
or 64
or 32
24
or 16
23
or 8
22
or 4
21
or 2
20
or 1
10011001 shown as
1
0
0
1
1
0
0
1
= 153
00101011 shown as
0
0
1
0
1
0
1
1
= 43
Figure 1-34. Binary system.
Place Value
10110011 shown as
27
or 128
26
or 64
25
or 32
24
or 16
23
or 8
22
or 4
21
or 2
20
or 1
1
0
1
1
0
0
1
1
128
+
0
+
32
+
16
+
0
+
0
+
2
+
1
= 179
Figure 1-35. Conversion from binary number to decimal number.
Place Value
27
26
25
or 128
or 64
or 32
24
or 16
23
or 8
22
or 4
21
or 2
20
or 1
35 shown as
0
0
1
0
0
0
0
1
124 shown as
0
1
1
1
1
1
0
0
96 shown as
0
1
1
0
0
0
0
0
255 shown as
1
1
1
1
1
1
1
1
233 shown as
1
1
1
0
1
0
0
1
Figure 1-36. Conversion from decimal number to binary number.
Converting Decimal Numbers to Binary Numbers
To convert a decimal number to a binary number, the
place values in the binary system are used to create a
sum of numbers that equal the value of the decimal
number being converted. Start with the largest binary
place value and subtract from the decimal number.
Continue this process until all of the binary digits are
determined.
Example: Convert the decimal number 233 to a binary
number.
Start by subtracting 128 (the largest place value from
the 8-bit binary number) from 233.
233 – 128 = 105
A “1” is placed in the first binary digit space:
1XXXXXXX.
Continue the process of subtracting the binary number
place values:
105 – 64 = 41
A “1” is placed in the second binary digit space:
11XXXXXX.
41 – 32 = 9
A “1” is placed in the third binary digit space:
111XXXXX.
Since 9 is less than 16 (the next binary place value),
a “0” is placed in the fourth binary digit space:
1110XXXX.
9–8=1
A “1” is placed in the fifth binary digit space:
11101XXX
Since 1 is less than 4 (the next binary place value), a 0
is placed in the sixth binary digit space: 111010XX.
Since 1 is less than 2 (the next binary place value), a 0 is
placed in the seventh binary digit space: 1110100X.
1–1=0
A “1” is placed in the eighth binary digit space:
11101001.
The decimal number 233 is equivalent to the binary
number 11101001, as shown in Figure 1-36.
Additional decimal number to binary number conversions are shown in Figure 1-36.
1-27
Conversion Factors
Length
Area
1 inch
2.54 centimeters
25.4 millimeters
1 square inch
6.45 square centimeters
1 foot
12 inches
30.48 centimeters
1 square foot
144 square inches
0.093 square meters
0.836 square meters
1 yard
3 feet
0.9144 meters
1 square yard
9 square feet
1 mile
5,280 feet
1,760 yards
1 acre
43,560 square feet
1 millimeter
0.0394 inches
1 square mile
640 acres
1 kilometer
0.62 miles
1 square centimeter
0.155 square inches
1 square meter
1.195 square yards
1 square kilometer
0.384 square miles
2.59 square kilometers
Volume
1 fluid ounce
29.57 cubic centimeters
1 cup
8 fluid ounce
1 pint
2 cups
16 fluid ounces
0.473 liters
1 quart
2 pints
4 cups
32 fluid ounces
0.9463 liters
1 gallon
4 quarts
8 pints
16 cups
128 ounces
1 gallon
231 cubic inches
1 liter
0.264 gallons
1 cubic foot
1,728 cubic inches
1 cubic foot
7.5 gallons
1 cubic yard
27 cubic feet
1 board foot
1.057 quarts
1 inch by 12 inches by 12 inches
Weight
1 ounce
28.350 grams
1 pound
16 ounces
1 ton
2,000 pounds
1 milligram
0.001 grams
1 kilogram
1,000 grams
1 gram
0.0353 ounces
Temperature
453.592 grams
0.4536 kilograms
degrees Fahrenheit to
degrees Celsius
Celsius = 5⁄9 × (degrees Fahrenheit − 32)
degrees Celsius to
degrees Fahrenheit
Fahrenheit = 9⁄5 × (degrees Celsius) + 32
2.2 pounds
Formulas for Area of Two-Dimensional Objects
Object
Area
Formula
Figure
Rectangle
Length × Width
A=L×W
1-13
Square
Length × Width or
Side × Side
A = L × W or
A = S2
1-14
Triangle
½ × (Length × Height) or ½
× (Base × Height)
or (Base × Height) ÷ 2
A = ½ (L × H) or
A = ½ (B × H) or
A = (B × H) ÷ 2
1-15
Parallelogram
Length × Height
A=L×H
1-17
Trapezoid
½ (base1 + base2) x Height
A = ½ (b1 + b2) x H
1-18
Circle
π × radius2
A = π × r2
1-20
Figure 1-37. Applied Mathematics Formula Sheet.
1-28
3.785 liters
Order of Operation for Algebraic Equations
Circumference of a Circle
1. Parentheses
2. Exponents
3. Multiplication and Division
4. Addition and Subtraction
Use the acronym PEMDAS to remember the order
of operation in algebra. PEMDAS is an acronym for
parentheses, exponents, multiplication, division, addition, and subtraction. To remember it, many use the
sentence, “Please Excuse My Dear Aunt Sally.”
Trigonometric Equations
Circumference of an Ellipse
Sine (sin) of angle A =
opposite side (side a)
hypotenuse (side c)
Cosine (cos) of angle A =
adjacent side (side b)
hypotenuse (side c)
opposite side (side a)
Tangent (tan) of angle A = adjacent side (side b)
Pythagorean Theorem
Formulas for Surface Area and Volume of
Three-Dimensional Solids
Volume
Surface Area
Figure
Rectangular
Solid
Solid
L×W×H
2 × [(W × L) + (W × H) +
(L × H)]
1-23
Cube
S3
6 × S2
1-24
2 × π × r2 + π × D × H
1-25
Sphere
4⁄ × π × r3
3
4 × π × r2
1-27
Cone
1⁄ × π × r2 × H
3
π × r × [r + (r2 + H2)1⁄2]
1-28
Cylinder
π ×
r2 × H
Figure 1-37. Applied Mathematics Formula Sheet. (continued)
1-29
Names and Symbols for Metric Prefixes
Powers of Ten
tera (1012)
One trillion times
Powers
of Ten
giga (109)
One billion times
Positive
Exponents
mega (106)
One million times
106
10 × 10 × 10 × 10 × 10 × 10
1,000,000
kilo (103)
One thousand times
105
10 × 10 × 10 × 10 × 10
100,000
104
10 × 10 × 10 × 10
10,000
103
10 × 10 × 10
1,000
102
10 × 10
100
10
10
Prefix
Means
Expansion
hecto (102)
One hundred times
deca (101)
Ten times
deci (10-1)
One tenth of
101
One hundredth of
100
milli (10-3)
One thousandth of
Negative
Exponents
micro (10-6)
One millionth of
10-1
1⁄10
1⁄10 = 0.1
One billionth of
10-2
1⁄(10 × 10)
1⁄100 = 0.01
10-3
1⁄(10 × 10 × 10)
1⁄1,000 = 0.001
10-4
1⁄(10 × 10 × 10 × 10)
1⁄10,000 = 0.0001
10-5
1⁄(10 × 10 × 10 × 10 × 10)
1⁄100,000 = 0.00001
10-6
1⁄(10 × 10 × 10 × 10 × 10 × 10)
1⁄1,000,000 = 0.000001
centi (10-2)
nano (10-9)
pico (10-12)
One trillionth of
1
Figure 1-37. Applied Mathematics Formula Sheet. (continued)
1-30
Value